# Quasi-periodic solutions of nonlinear beam equation with prescribed frequencies

## Abstract

Consider the one dimensional nonlinear beam equation u{sub tt} + u{sub xxxx} + mu + u{sup 3} = 0 under Dirichlet boundary conditions. We show that for any m > 0 but a set of small Lebesgue measure, the above equation admits a family of small-amplitude quasi-periodic solutions with n-dimensional Diophantine frequencies. These Diophantine frequencies are the small dilation of a prescribed Diophantine vector. The proofs are based on an infinite dimensional Kolmogorov-Arnold-Moser iteration procedure and a partial Birkhoff normal form. .

- Authors:

- College of Information Technology, Jilin Agricultural University, Changchun 130118 (China)
- School of Mathematics and Statistics, and Center for Mathematics and Interdisciplinary Sciences, Northeast Normal University, Changchun 130024 (China)

- Publication Date:

- OSTI Identifier:
- 22403146

- Resource Type:
- Journal Article

- Journal Name:
- Journal of Mathematical Physics

- Additional Journal Information:
- Journal Volume: 56; Journal Issue: 5; Other Information: (c) 2015 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0022-2488

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; BEAMS; DIRICHLET PROBLEM; EQUATIONS; MATHEMATICAL SOLUTIONS; NONLINEAR PROBLEMS; ONE-DIMENSIONAL CALCULATIONS; PERIODICITY

### Citation Formats

```
Chang, Jing, Gao, Yixian, E-mail: gaoyx643@nenu.edu.cn, and Li, Yong.
```*Quasi-periodic solutions of nonlinear beam equation with prescribed frequencies*. United States: N. p., 2015.
Web. doi:10.1063/1.4919673.

```
Chang, Jing, Gao, Yixian, E-mail: gaoyx643@nenu.edu.cn, & Li, Yong.
```*Quasi-periodic solutions of nonlinear beam equation with prescribed frequencies*. United States. doi:10.1063/1.4919673.

```
Chang, Jing, Gao, Yixian, E-mail: gaoyx643@nenu.edu.cn, and Li, Yong. Fri .
"Quasi-periodic solutions of nonlinear beam equation with prescribed frequencies". United States. doi:10.1063/1.4919673.
```

```
@article{osti_22403146,
```

title = {Quasi-periodic solutions of nonlinear beam equation with prescribed frequencies},

author = {Chang, Jing and Gao, Yixian, E-mail: gaoyx643@nenu.edu.cn and Li, Yong},

abstractNote = {Consider the one dimensional nonlinear beam equation u{sub tt} + u{sub xxxx} + mu + u{sup 3} = 0 under Dirichlet boundary conditions. We show that for any m > 0 but a set of small Lebesgue measure, the above equation admits a family of small-amplitude quasi-periodic solutions with n-dimensional Diophantine frequencies. These Diophantine frequencies are the small dilation of a prescribed Diophantine vector. The proofs are based on an infinite dimensional Kolmogorov-Arnold-Moser iteration procedure and a partial Birkhoff normal form. .},

doi = {10.1063/1.4919673},

journal = {Journal of Mathematical Physics},

issn = {0022-2488},

number = 5,

volume = 56,

place = {United States},

year = {2015},

month = {5}

}

DOI: 10.1063/1.4919673

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